Circuits - 2

OK, so about regular circuits:

The images represent SERIES CIRCUITS and PARALLEL CIRCUITS.

In a series circuit, the current is constant and is set by the total resistance of the circuit (the sum of the resistors). If you remove one resistor (or light bulb, as in the first image), the current stops. If the resistors were identical bulbs, having more bulbs would result in dimmer bulbs, since the battery voltage is distributed among them.  Note that the sum of the voltages "over" the bulbs is equal to the total voltage provided by the battery (give or take some minor losses).  Identical bulbs (or resistors) have identical voltages "over" them - 3 identical bulbs connected to a 9-V battery would have roughly 3-V each over them.

In parallel circuits, current has multiple paths to take, so the total resistance of the circuit is actually LESS than if the resistors were alone or in series with other resistors - see details below. Since the bulbs are connected equally to the battery, they experience the same as the battery voltage - they are, therefore, of equal brightness (and the same brightness they would have if there were only ONE bulb connected). Of course, bulbs in parallel draw more current and thus cause a battery to die sooner.  You could have 10 bulbs or resistors connected in parallel to a battery - each will be as bright as if only 1 were connected to the battery (same voltage over each), though 10 bulbs will kill the battery 10 times faster.

Does this have anything to do with holiday lights?

What I've written above is primarily geared toward identical bulbs. In series, add up the resistances to get the total resistance. In parallel, it is more complicated. There is a formula one can use (1/Rp = 1/R1 + 1/R2 + ...), but we will only concern ourselves with the case of identical resistors in parallel. In that case, divide the value of the resistor by the number of resistors to get the total effective resistance. For example, two identical 50-ohm resistors in parallel is the same as one 25-ohm resistor. This seems strange, but it's a little like toll booths - when one toll booth is open, it can get crowded (the current is small). With multiple toll booths open, the resistance is effectively less, so the current can be greater.

In the first image below, the graphic represents the schematic view of a parallel circuits, with 2 resistors.  Note that 2 possible paths are available for current to take - current runs through EACH path, though there will be more current where there is less resistance.  The total current from the battery is equal to the sum of the currents through the 2 resistors.  It follows V = I R, though the V over each R is the same.  The I through each will therefore be V/R.

The second image illustrates the series circuit concept:  identical resistors in series will effectively give MORE resistance (the sum of the resistances, actually) to the battery, so the current will be LESS (and exactly the same in each resistor or bulb).  It also easily follows V = I R, with more R yielding less I (when V is constant).  Think of V = I R this way:  I = V/R.  More R, less I.

Circuits - 1

Thus far, we have only discussed "static" (stationary) charges.  Static charges alone are useful, but not nearly as much as charges in motion.  As you recall, electrons are the most easily moved particles.  However, for sake of ease in sign convention (positive vs. negative), we define the following:

Current (I) - the rate at which positive charge "flows"

I = Q/t

The unit is the coulomb per second, defined as an ampere (A).  One ampere (or amp) is a tremendous amount of current - more than enough to kill a person.  In fact, you can feel as little as 0.01 A.  Typical currents in a circuit are on the order of mA (milliamperes).

Essentially, current is how quickly charge travels (or charge per time, q/t).  The unit (a coulomb per second) is called the ampere (or amp, A). 

We need to define other new quantities in electricity:  voltage, resistance, power.

Voltage (V) - the amount of available energy per coulomb of charge.  The unit is the joule per coulomb, called a volt (V, in honor of Allesandro Volta, inventor of the battery).

V = E/Q

Batteries* and other sources (such as wall sockets) "provide" voltage, which is really a difference between TWO points (marked + and - on a battery).  A wall outlet is a bit more complex - there are 2 prongs, but often also a third prong (the "ground", for safety purposes, through which excess charge can travel back to the Earth).

Or schematically:

Note the symbols for battery, lamp, wire (or conductor), and switch.  All electrical components have symbols that are internationally recognized.

Resistance (R) - the ratio of voltage applied to an electrical device to the current that results through the device.  Alternately:  the amount by which the voltage is "dropped" per ampere of current.

R = V/I

You can also think of resistance as that which "resists" current.  Typically, resistors are made of things that are semi-conductive (they conduct current, but less well than conductors and better than insulators).  Resistors are often made of carbon, but can also be made of silicon and other materials.  The unit is the volt per ampere, defined as an ohm (Greek symbol omega)

A convenient way to relate all of the variables is embodied in an expression often called Ohm's Law:

V = I R

And the schematic symbols for resistor are:

Or:

So, what exactly IS a circuit?

An electrical circuit can be thought of as a complete "loop" through which charge can travel.  Therefore, it actually has to be physically complete - there can be no openings.  That is, the current actually has to have a complete path to take.  I will demonstrate this in class with bulbs and wires; for now, see the image above.

https://phet.colorado.edu/en/simulation/circuit-construction-kit-dc

Some folks like analogies.  Consider a water analogy.  Voltage is like a tank of water (how much water).  Resistance is provided by a drain or faucet.  The rate at which water comes out is the current.  It's only an analogy, but it gets the gist of circuit terminology ok.

* A bit more about batteries and other electrochemical cells.  The actual "voltage" produced is the result of 2 different metals immersed in a substance which reacts with them.  The metals differ in the extent to which they "give up" electrons, so electrons can "flow" from one metal to the other - which is made MUCH easier by connecting the two with a conductor of some sort.  For more chemistry info, go here:

http://hyperphysics.phy-astr.gsu.edu/hbase/Chemical/electrochem.html

Charge questions

Things to think about:

1.  What exactly *is* charge?  How do we think of it?  How does this relate to protons and electrons, etc.?

2.  Why is it that electrons are the easiest particles to manipulate?

3.  What does atomic number (Hydrogen = 1, Helium = 2, etc.) mean?  Also - What does charging an atom entail - How can an atom be charged negatively or positively?

4.  What are quarks?

5.  Coulomb's law is an "inverse square law" - what does this mean exactly?

6.  Why can a charged balloon stick to a wall?

7.  What is "grounding"?

8.  Recall the demonstration where I charged up the small suspended piece - what was I showing?  Or the demo with the pie plates.

9.  Is the charge of an electron a big or small number?  Roughly, how many electrons would be needed to make 1 coulomb of charge?

Electricity Part 1 - Charge!

Charge

- as fundamental to electricity & magnetism as mass is to mechanics

Charge is a concept used to quantatively related "particles" to other particles, in terms of how they affect each other - do they attract or repel?  If so, with what force?

Charge is represented by letter Q.

The basic idea - likes charges repel (- and -, or + and +) and opposite charges attract (+ and -).

Charge is measured in units called coulombs (C).  A coulomb is a huge amount of charge, but a typical particle has a tiny amount of charge:

- the charge of a proton is 1.6 x 10^-19 C.  Similarly, the charge of an electron is the same number, but negative, by definition (-1.6 x 10^-19 C).  The negative sign distinguishes particles from each other, in terms of whether or not they will attract or repel.  The actual sign is arbitrarily chosen.

The charge of a neutron is 0 C, or neutral.


But what IS charge?

Charge is difficult to define.  It is property of particles that describes how particles interact with other particles. 

In general, the terms are negative and positive, with differing amounts of each, quantified as some multiple of the fundamental charge value (e):

e = 1.6 x 10^-19 C

That's hard to visualize, since a coulomb (c) is a huge amount of charge.  One coulomb, for example, is the charge due to:

1 coulomb = charge due to 6.3 x 10^18 protons

A typical cloud prior to lightning may have a few hundred coulombs of charge - that's an enormous amount of excess charge.

If the charge is negative (-), the excess charge is electrons.

If the charge is positive (+), the excess charge is protons - however, we can NOT easily move protons.  That usually takes a particle accelerator.  Typically, things are charged positively by REMOVING electrons, leaving a net charge of positive.

Other things to remember:

Neutral matter contains an equal number of protons and electrons.

The nucleus of any atom contains protons and (usually) neutrons (which carry no charge).  The number of protons in the nucleus is called the atomic number, and it defines the element (H = 1, He = 2, Li = 3).

Electrons "travel" around the nucleus in "orbitals."  See chemistry for details.  The bulk of the atom is empty space.

Like types of charge repel.  Opposite types of charge attract.

The proton is around 2000 times the mass of the electron and makes up (with the neutrons) the bulk of the atom.  This mass difference also explains why the electron orbits the proton, and not the other way around.

Protons in the nucleus of an atom should, one would imagine, repel each other greatly.  As it happens, the nucleus of an atom is held together by the strong nuclear force (particles which are spring-like, called gluons, keep it together).  This also provides what chemists called binding energy, which can be released in nuclear reactions.

COULOMB'S LAW

How particles interact with each other is governed by a physical relationship called Coulomb's Law:

F = k Q1 Q2 / d^2

Or, the force (of attraction or repulsion) is given by a physical constant times the product of the charges, divided by their distance of separation squared.  The proportionality constant (k) is used to make the units work out to measurable amounts.

Note that this is an inverse square relationship, just like gravity.

The "big 3" particles you've heard of are:

proton
neutron
electron

However, only 1 of these (the electron) is "fundamental".  The others are made of fundamental particles called "quarks""

proton = 2 "up quarks" + 1 "down quark"
neutron = 2 "down quarks" + 1 "up quark"

There are actually 6 types of quarks:  up, down, charm, strange, top, & bottom.  The names mean nothing.

Many particles exist, but few are fundamental - incapable of being broken up further.

In addition, "force-carrying" particles called "bosons" exist -- photons, gluons, W and Z particles.

The Standard Model of Particles and Interactions:

http://www.pha.jhu.edu/~dfehling/particle.gif

LENSES!

Lenses

As shown and discussed in class, light refracts TOWARD a normal line (dotted line on the left image, perpendicular to surface of lens) when entering a more dense medium.

Note in this convex lens that this direction of bend changes from down (with the top ray) to up with the bottom ray. This is due to the geometry of the lens. Look at the picture to make sure that this makes sense.  As a result, the rays will intersect after leaving the lens.  An image can form!


The FOCAL LENGTH (f) of a lens (or curved mirror) where the light rays would intersect, but ONLY IF THEY WERE INITIALLY PARALLEL to each other. Otherwise, they intersect at some other point, or maybe not at all (if the object is too close to be focused on)!

Note that your (human) eye lenses are convex - slightly thicker in the middle.  Thus, your eyes form "real" images on the retina - upside-down!  Unless, of course, the object is too close.

If an image is projected onto a screen, the image is REAL. Convex lenses (fatter in the middle) CAN create real images - the only cases where there are no images for convex lenses are when the object distance (between object and lens) is equal to the f, or when do < f. In the first case, there is NO image at all. In the second case, there is a magnified upright virtual image "inside" the lens.

Concave lenses (thinner in the middle) NEVER create real images and ONLY/ALWAYS create virtual images.

Top image depicts parallel light rays hitting a convex lens and meeting at the "focal point."  A real image forms at the focal length of a convex lens, WHEN THE RAYS ARE INITIALLY PARALLEL.  People who are farsighted wear convex lenses.

The bottom image depicts parallel light rays hitting a concave lens and diverging.  In this case, under all circumstances (regardless of where the object is), only virtual images are formed.  These can not be projected onto a screen - rather, they appear to reside "inside" the lens.  People who are nearsighted wear concave lenses.




However, unless the light rays are exactly parallel (or the object is so far away, like the Sun, so that they are approximately parallel), the light rays do not behave exactly like this.  Rather, they form at a different location.

Extension to curved mirrors:

Convex lenses (which are defined to have a positive focal length) are similar to concave mirrors.

Concave lenses (which are defined to have a negative focal length) are similar to convex mirrors.


Summary

The key thing to note is that whether or not an image forms, and what characteristics that image has, depends on:

- type of lens or mirror
- how far from the lens or mirror the object is

In general, convex lenses (and concave mirrors) CAN form "real" images.  In fact, they always form real images (images that can be projected onto screens) if the object is further away from the lens/mirror than the focal length.   Think of using a magnifying glass to burn leaves - a real image of the Sun is forming on the leaves.

If the object is AT the focal point, NO image will form.

If the object is WITHIN the focal point (less than the focal point), only virtual images (larger ones) will form "inside" the mirror or lens.

Concave lenses and convex mirrors ONLY form virtual images; they NEVER form real images.  Think of convenience store mirrors and glasses for people who are nearsighted.

Extra info, FYI:

The location of images can be predicted by a powerful equation:

1/f = 1/di + 1/do

In this equation, f is the theoretical focal length (determined by the geometry of the lens or mirror), do is the distance between the object and lens (or mirror) and di is the distance from lens (or mirror) to the formed image.

We find several things to be true when experimenting with lenses. If the object distance (do) is:

greater than 2f -- the image is smaller
equal to 2f -- the image is the same size as the object (and is located at a di equal to 2f)
between f and 2f -- the images is larger
at f -- there is NO image
within f -- the image is VIRTUAL (meaning that it can not be projected onto a screen) and it appears to be within the lens (or mirror) itself

Optics applets/animation

Refraction, in general:

http://www.absorblearning.com/media/item.action?quick=15n

https://phet.colorado.edu/en/simulation/bending-light

http://www.physicsclassroom.com/Physics-Interactives/Refraction-and-Lenses/Refraction/Refraction-Interactive

And for lenses:

https://phet.colorado.edu/en/simulation/geometric-optics

http://courses.cs.washington.edu/courses/cse131/12au/applets/thinlens.html

Recall also what happens when light hits mirrors (flat and curved).

Exam 2 review

Exam 2 topics:


Energy
Basics of flight; Bernoulli effect

waves

- wavelength

- frequency

- speed

-amplitude

- crests and troughs

wave speed = frequency x wavelength

(Note that the wave speed is the speed of light when you are talking about electromagnetic waves.)

mechanical vs. electromagnetic waves

harmonics on a string - "standing waves"

music - octaves, the next note on the piano (1.0594)

Doppler effect

- red shift, blue shift

light reflection

light refraction

lenses and mirrors (convex and concave)

focal length/point

predicting light paths (when light is reflected or refracted)

electromagnetic spectrum - radio, micro, IR, visible, UV, X, gamma

Optics questions

Optics questions - answers forthcoming

1.  Review the concept of reflection, particularly the law of reflection.  Draw what happens when a light ray hits a mirror at various angles.

2.  Review the concept of refraction:  what it is, what causes it, what happens during it, under what circumstances does light bend, etc.  Draw what happens when a light ray hits a block of transparent plastic at various angles.  

3.  Show how to calculate the wavelength of WTMD's signal (89.7 MHz).

4.  Some questions related to how light is affected by optics.

Answers:

Light - 3. Refraction

Refraction:

Consider a wave hitting a new medium - one in which is travels more slowly. This would be like light going from air into water. The light has a certain frequency (which is unchangeable, since its set by whatever atomic process causes it to be emitted). The wavelength has a certain amount set by the equation, c = f l, where l is the wavelength (Greek symbol, lambda).

When the wave enters the new medium it is slowed - the speed becomes lower, but the frequency is fixed. Therefore, the wavelength becomes smaller (in a more dense medium).

Note also that the wave becomes "bent." Look at the image above: in order for the wave front to stay together, part of the wave front is slowed before the remaining part of it hits the surface. This necessarily results in a bend.

MORE DETAIL:

The general rule - if a wave is going from a lower density medium to one of higher density, the wave is refracted TOWARD the normal (perpendicular to surface) line. See picture above.

Refraction is much different than reflection. In refraction, light enters a NEW medium. In the new medium, the speed changes. We define the extent to which this new medium changes the speed by a simple ratio, the index of refraction:

n = c/v

In this equation, n is the index of refraction (a number always 1 or greater), c is the speed of light (in a vacuum) and v is the speed of light in the new medium.

The index of refraction for some familiar substances:

vacuum, defined as 1

air, approximately 1

water, 1.33

glass, 1.5

polycarbonate ("high index" lenses), 1.67

diamond, 2.2

The index of refraction is a way of expressing how optically dense a medium is. The actual index of refraction (other than in a vacuum) depends on the incoming wavelength. Different wavelengths have slightly different speeds in (non-vacuum) mediums. For example, red slows down by a certain amount, but violet slows down by a slightly lower amount - meaning that red light goes through a material (glass, for example) a bit faster than violet light. Red light exits first.

In addition, different wavelengths of light are "bent" by slightly different amounts. This is trickier to see, but it causes rainbows and prismatic effects.

Some animation, etc.:

http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html

http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm

http://www.freezeray.com/flashFiles/Refraction2.htm



And all of this helps explain how lenses form images.

Light - 2. Reflection.

Reflection - light "bouncing" off a reflective surface. This obeys a simple law, the law of reflection!

The incident (incoming) angle equals the reflected angle. Angles are generally measured with respect to a "normal" line (line perpendicular to the surface).

Note that this works for curved mirrors as well, though we must think of a the surface as a series of flat surfaces - in this way, we can see that the light can reflect in a different direction, depending on where it hits the surface of the curved mirror.

So - light reflects from mirrors, according to the law of reflection.  However, if the mirrors is curved, light still obeys this rule - it just looks a bit different.  You have to visualize the curved mirror as a series of little flat mirrors.

A convex mirror (top) acts reflects light rays "outward" - the light rays seem as though they are coming from inside the convex mirror, so it seems as though there is an image inside.  We call this a VIRTUAL IMAGE.  Think of convenience store mirrors or side view mirrors.

 A concave mirror (bottom) acts sort of the opposite way.  The parallel light rays bend "inward" - so the light rays converge at a FOCAL POINT.  Where they meet, an image is formed - we call this a REAL IMAGE.

Note however that this happens in this case because the light rays were initially parallel (which is what happens if the object the light rays reflect from is far away).  If they are NOT initially parallel - in other words, if the object is reasonably close to the mirror, the rays may converge at some other point.  Examples of concave mirrors are found in makeup/shaving mirrors and reflecting telescope mirrors.  But again - the light rays ONLY meet at the focal point IF they were initially parallel.  If not, they meet elsewhere (or maybe not at all).  More about this next class when we talk about refraction and lenses.

Light - 1

Recall that waves can be categorized into two major divisions:

Mechanical waves, which require a medium. These include sound, water and waves on a (guitar, etc.) string

Electromagnetic waves, which travel best where there is NO medium (vacuum), though they can typically travel through a medium as well. All electromagnetic waves can be represented on a chart, usually going from low frequency (radio waves) to high frequency (gamma rays). This translates to: long wavelength to short wavelength.

All of these EM waves travel at the same speed in a vacuum: the speed of light (c). Thus, the standard wave velocity equation becomes:

c = f l

where c is the speed of light (3 x 10^8 m/s), f is frequency (in Hz) and l (which should actually be the Greek letter, lambda) is wavelength (in m).

General breakdown of e/m waves from low frequency (and long wavelength) to high frequency (and short wavelength):

Radio

Microwave

IR (infrared)

Visible (ROYGBV)

UV (ultraviolet)

X-rays

Gamma rays

In detail, particularly the last image:

http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html

http://www.unihedron.com/projects/spectrum/downloads/full_spectrum.jpg

Don't forget - electromagnetic waves should be distinguished from mechanical waves (sound, water, earthquakes, strings on a guitar/piano/etc.). 

ALL E/M waves (in a vacuum) travel at the SPEED OF LIGHT (c).

Some questions for practice.

Related questions from today's class on the Doppler effect:

1.  What is the Doppler effect?

2.  An ambulance is coming toward you, with a siren blasting a 1000 Hz tone.
a.  As it approaches you, what will be true of the frequency of the sound YOU hear?
b.  After it passes you, what will be true of the frequency of the sound YOU hear?
c.  What is true of the frequency of sound that the ambulance drivers hear?

2.  Be sure to review the websites shown in class.  Remember that the Doppler effect has to do with frequencies, NOT volumes of waves.

3.  What is a red shift?  What is a blue shift?


Also - questions from earlier material (before we did waves):

1.  What is the Bernoulli principle?

2.  Explain the general principle behind why airplanes fly.

3.  In general, what is energy?